Piecewise Functions: Graphing Made Easy

Algebra 2 Grades High School 9:36 Video
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Lesson Description

Learn how to graph piecewise functions using two easy methods: the 'graph and erase' method and the 'table' method. This lesson provides a step-by-step guide with examples to master graphing piecewise functions.

Video Resource

Graphing Piecewise Functions (2 Easy Methods)

Mario's Math Tutoring

Duration: 9:36
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Key Concepts

  • Piecewise functions are functions defined by different expressions over different intervals of their domain.
  • The 'graph and erase' method involves graphing each piece of the function over the entire domain and then erasing the portions that do not belong to the specified interval.
  • The 'table' method involves creating a table of values for each piece of the function, focusing on values within the specified interval.

Learning Objectives

  • Students will be able to define piecewise functions and identify their key components.
  • Students will be able to graph piecewise functions using the 'graph and erase' method.
  • Students will be able to graph piecewise functions using the 'table' method.
  • Students will be able to determine the domain and range of a piecewise function from its graph.

Educator Instructions

  • Introduction (5 mins)
    Begin by defining piecewise functions and explaining their structure. Emphasize that different 'pieces' of the function apply to different intervals of the x-axis. Briefly introduce the two graphing methods that will be taught.
  • Graph and Erase Method (15 mins)
    Explain the 'graph and erase' method. Demonstrate how to graph each piece of the function as if it were defined over the entire domain. Then, show how to erase the portions of each graph that fall outside the specified interval. Emphasize the importance of using open or closed circles to indicate whether endpoints are included or excluded.
  • Table Method (15 mins)
    Explain the 'table' method. Demonstrate how to create a table of x and y values for each piece of the function, focusing on values within the specified interval. Emphasize the importance of including endpoints in the table and using open or closed circles based on the inequality symbols. Plot the points from each table and connect them to create the graph.
  • Examples and Practice (15 mins)
    Work through several examples of graphing piecewise functions using both methods. Encourage students to try graphing the functions themselves before revealing the solutions. Point out when one method might be easier or more efficient than the other. Address any questions or misconceptions.
  • Conclusion (5 mins)
    Summarize the key concepts of piecewise functions and the two graphing methods. Reiterate the importance of understanding the domain restrictions for each piece of the function. Assign practice problems for homework.

Interactive Exercises

  • Graphing Challenge
    Present students with a set of piecewise functions and challenge them to graph each function using either the 'graph and erase' or 'table' method. Encourage them to compare their results with classmates and discuss any discrepancies.

Discussion Questions

  • What is a piecewise function, and how does it differ from other types of functions?
  • When might the 'graph and erase' method be more efficient than the 'table' method, and vice versa?
  • How do you determine whether to use an open or closed circle at the endpoint of a piece of a piecewise function?

Skills Developed

  • Graphing functions
  • Understanding domain and range
  • Problem-solving
  • Analytical thinking

Multiple Choice Questions

Question 1:

Which of the following best describes a piecewise function?

Correct Answer: A function defined by multiple sub-functions, each applying to a specific interval of the domain.

Question 2:

In the 'graph and erase' method, what do you do after graphing each piece of the function?

Correct Answer: Erase the parts of the graph that are outside the specified domain interval for that piece.

Question 3:

What does an open circle on the graph of a piecewise function indicate?

Correct Answer: The point is excluded from the function's domain.

Question 4:

When creating a table for a piecewise function, which values should you include?

Correct Answer: Only values within the specified interval.

Question 5:

Which of the following functions is a piecewise function?

Correct Answer: f(x) = |x|

Question 6:

What is the purpose of considering the domain restrictions when graphing piecewise functions?

Correct Answer: To ensure each part is graphed correctly only where defined.

Question 7:

Which method involves plotting individual points to create each piece of the function?

Correct Answer: Table Method

Question 8:

What is the first step in graphing with the "graph and erase" method?

Correct Answer: Graph each piece of the function as if the domain was not restricted.

Question 9:

What type of circle is used at x=2 when the domain restriction is x<2?

Correct Answer: Open Circle

Question 10:

What type of circle is used at x=2 when the domain restriction is x>=2?

Correct Answer: Closed Circle

Fill in the Blank Questions

Question 1:

A __________ function is a function defined by different expressions over different intervals of its domain.

Correct Answer: piecewise

Question 2:

The 'graph and erase' method involves graphing each piece and then __________ the parts that are not in the specified domain.

Correct Answer: erasing

Question 3:

An __________ circle indicates that a point is not included in the domain.

Correct Answer: open

Question 4:

The __________ method involves creating a table of x and y values for each piece of the function.

Correct Answer: table

Question 5:

When using the 'table' method, you should pay special attention to the __________ of each interval.

Correct Answer: endpoints

Question 6:

A __________ circle indicates that a point is included in the domain.

Correct Answer: closed

Question 7:

When x is greater than or equal to a value, the domain includes all numbers to the __________ of that value on a number line.

Correct Answer: right

Question 8:

When x is less than or equal to a value, the domain includes all numbers to the __________ of that value on a number line.

Correct Answer: left

Question 9:

If f(x) = 2x for x<0 and f(x) = x+1 for x>=0, then f(-1) is equal to __________.

Correct Answer: -2

Question 10:

A piecewise function is still a _________ as long as it passes the vertical line test.

Correct Answer: function

Educational Standards

A2.F.1:Understand functions as descriptions of covariation (how related quantities vary together). A2.F.1.1:Use algebraic, interval, and set notations to specify the domain and range of various types of functions, and evaluate a function at a given point in its domain. A2.F.1.2:Identify the parent forms of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [𝑓(𝑥 + 𝑐), 𝑓(𝑥) + 𝑐, 𝑓(𝑐𝑥), and 𝑐𝑓(𝑥)] algebraically and graphically.

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